Do philosophers distinguish between mathematical creativity, and the broader artistic creativity? If so, what are the differences between these two?
A lot of people seem to treat IQ as something unrelated to creativity when it seems to be the case for artistic creativity, but not mathematical creativity.
There are caricatures of math and the arts, and then there are characterizations, the best of which are accurate. Many students get dragged through the drudgery of mathematical algorithms and washout without ever seeing the beauty and creativity inherent in the discipline, so let's be a little more thorough and explore what philosophers have discovered about creativity. I'd argue that philosophers of mathematics are engaged in an entirely creative act like any sculptor is. The only difference is the medium.
In traditional concepts of mathematics and art, there are different underlying objects being studied. Mathematics focuses on the logic, direction, shape, quantity, and relations expressed in language, whereas art (which is a very broad term which has fine and commercial varieties) tends to refer to the novel expression of auditory and visual media, be it literature, poetry, paints, sculpture, music, and dance. In fact, there can be a lot of overlap depending on the interests of the thinker. I have a geometry textbook that relies heavily on the works of M.C. Escher to illustrate lessons. Is geometric tiling mathematics or art? Obviously both, with the difference arising from the methodology and aims of thinker. You say:
a lot of people seem to treat IQ as something unrelated to creativity
Many people do not see mathematics and logic as a creative venture, but I suspect that is more about how human beings are socialized (math drill! one-correct-logic!) than practicing. In philosophy, for instance, Imre Lakatos inspired by Karl Popper, someone famous for railing against scientific consensus regarding methodology, insisted mathematics was a creative endeavor. Certainly, if you read Polya's How to Solve It (GB), the text claims mathematics is a heuristic rather than algorithmic endeavor. This, in fact, is a common theme among the mathematically and scientifically inclined starting since the 1960's and the failure of the program of logical positivism. Paul Feyerabend wrote a book Against Method because he saw that at the fundamental core of thought, there is no algorithm, only heuristic. (A very important mathematical distinction.)
Having a high IQ is often a sign of seeing things in quickly and in a novel way. The Mensa entrance examination doesn't present anything in the way of conceptual challenge, but rather in seeing a solution or pattern with an overwhelming problem space. One could argue that solving one of Raven's Progressive Matrices is an act of creativity, because the human solves it generally by a flash of insight rather than a calculation. And if you plumb the depths of mathematical genius, you'll find that the great names created math, they didn't just follow a recipe to get to a solution. Nikolai Lobachevsky is immortal in math history because he created an alternative to the axioms of Euclid, not necessarily because he mastered them. Great math is not solving a calculus problem to pass a test, it's envisioning the fluxion or differential and creating the calculus to begin with. Any professional mathematician will tell you that they are engaged in a creative act.
The real question is, are the mechanisms that allow for creativity in the fine and commercial arts the same that are used in mathematics? If you believe in a psychometrician's claim about G factor, the answer is broadly, yes. Honestly, there is some dispute on what characterizes intelligence, and the one commonality between the positions embodied by positions like Spearman's, Gardner's MI theory, or Cattell and Horn's views is the invocation of creativity which is often conceived of assembling ideas, words, or other primitives like color and sound, in novel ways. Hence, creativity, while domain specific, has a permutational flavor in all domains.
In my own domain of computer science, N vs. NP is a question which is open and bears on the question of creativity. In the study of computational theory, questions like the Halting problem have rigorous proofs in mathematical logic that suggest that there are some truths, some proofs, and thus some ideas that are inescapably beyond algorithmic methods and rely on intuitions. The notion of effective computability explored by Alan Turing, a great mathematician and logician who helped found computer science, constructed a concept called an oracle machine precisely because even in rigorous math and logic, sometimes the only way to get there is through creative insights. Goedel talks about unprovable truth, and modern philosophy of mind is built around defeasibility (SEP) and non-monotonic logic (SEP). One of my motivations for persisting on this site is to spread the word that mathematics and computer science are more than algorithms and machines that run them.
The fact that your visual system creates color and doesn't discover it should give you pause about how deeply the act of creativity extends. Modern science and some modern philosophy, in fact, supports the idea that the entirety of conscious thought and perhaps the world as we understand it is an act of creativity at some level of the brain (IEP).
Yes, in essence.
For my argument I will consider mathematicians only as creators of mathematical stuff, not just copiers and learners. After all we don't call proof-readers and copy-typists authors.
The common thing between mathematicians and artists is seeking of balance. A painting is beautiful if it has a balance of some kind. Even if its lop-sided in some way its balancing impact of previous lop-sided paintings.
A poetry is obviously balanced. It usually rhymes. It takes same mental load and same time to read the first line in a couplet as it takes to read the second one.
A mathematical equation is also ofcourse balanced.
Mathematics is all about operations. To display how an operation affects the operands there has to be an equation. Without equations you cannot show how an operation works. You have to write something like:
A x B = ...
An equation will always be there. Note that "A" and "B" can mean a lot of things depending on context. They can be simple natural numbers, complex numbers, matrices even vectors etc. The operator "x" can also mean different things, such as multiplication, cross product etc.
Anybody who don't agree with this have to provide just one example where a mathematical creativity is done outside context of any equation, or a piece of art thats not balanced in anyway.
Creativity can be defined as the ability of an individual to think divergently from other people to arrive at novel solutions to problems. In that sense, the concept is fundamentally the same between the fields of creative arts and mathematics, but the capacity itself may be completely distinct. The "problems" to be solved in these domains are radically different. At minimum for a person to be creative in a given field, they need to be good at solving the relevant problems.
The relationship between creativity and intelligence is primarily an empirical question which has been studied by experimental psychology. I'm not sure if this research has drilled down much into the two fields of art versus math. On the one hand, it suggests to me that intelligence is likely to be complementary in any field. On the other hand, intelligence as it is normally defined is more directly important for solving problems in mathematics then it is in the arts. In that sense I think your suggestion of "IQ as something unrelated to... artistic creativity, but not mathematical creativity" is too imprecise to be true or false.
No. To start, the difference must be remarked.
Epistemologically, every discipline covers three domains: science, technique and art. Science is the theoretical part, technique is applied science and art is socially applied technique (cf. Mario Bunge).
For example, knowing the musical theory without knowing how to play implies knowing only the empirical truths of music (that is science, precisely), but not being able to apply them. Same occurs when someone knows the theory of making shoes, without actually having the manual skills to make one. Notice that this domain is essentially rational, abstract.
Knowing how to play a guitar implies having a theoretical knowledge and a technique, but that is not enough to move anyone (plenty of guitar players are incredibly skilled, while their music is boring, even disgusting). An skilled shoemaker can make a shoe which is technically perfect, while nobody finds it useful or attractive. This domain is not rational, but empirical, practical, while is still abstract (the technique focuses the rules, not the results).
The last domain is the art, which is not only related with aesthetics, but to the social contribution a technique can make (art as in state of the arts). A shoemaker making a good shoe, that pleases others. A musician using its technique to move others' emotionally, etc. This domain is empirical/practical and concrete/particular.
So, the differences in creativity are evident: creativity in the theoretical domain (maths can be considered part of the theoretical domain) is essentially rational, abstract. Creativity in the artistic domain is practical, social, applied, having mostly concrete, particular features.
Are they the same? Let's see here. Can a massive-big-brain mathematician paint a portrait? Write poetry? Compose a symphony?
Not necessarily. It's not the way to expect things to go.
Try it the other way. Can a scary-good portrait artist do mathematics? Can a massively respected symphony composer do number theory? Can your favorite poet prove L'Hôpital's rule?
Not necessarily. This way isn't particularly likely either.
Are there aspects that overlap? Sure. Becoming good at something requires persistence, effort, at least some degree of intelligence, and so on. There is something we often call "inspiration" involved in each. And it is often done at a level that the initiated find difficult to access. The first time you hear a composition by Gage may leave you struggling to understand in much the same way you might the first time you encountered the equations of general relativity.
But they are not "the same."
Art relies on culture which is built on social practice. Differently said, culture belongs to people who identify themselves as part of the chosen culture by having behaviors that are commonly used or accepted from said culture.
Considering creativity as a part of freewill, because of the nature of freewill to highlight individuality, freewill is an identity's asset due to the abilities of freewill to assert identity by using creativity as a expression tool of identity and assuming that identity and individuality are bond together. Because of Math is defined as a tool able to resolve an issue by using his own science, and that the outcome from Math applied to an issue dont depend of freewill, so, freewill nully affect outcome of Math where creativity of freewill tend to assert identity and underlying culture.
The creative process doesn't serve the same purpose as one, that of the creation of art, which is driven by aesthetic motives and opinions where mathematical creativity follows an independent intellectual path of his development.
Lets says the issues is to fill a bottle with two glass, one green and one blue. You can put half of the green and then a quart of the blue or use another glass whatevers the sequence order or how many glass you want. At the end, you are going to have two empty glass and one bottle filled, at least. If your freewill comply to the idea:"im going to fill in my way", so, there is a freewill in Math. But the outcome and pragmatic effect is going to be the same whatever the path you have choosen.
I think a math teacher knows what a beautiful answer is.
Yes, one can display elegance and beauty in mathematics, and if you are serious in the subject you strive for beauty in results, approaches etc. I myself am average in the subject. I'm working so hard at it that I can't raise my work to a beautiful level.
